Semidefinite programs (SDPs) are a class of convex optimization problems whose domain is formed by the intersection of the cone of positive semidefinite (PSD) matrices with affine subspaces. Having high expressive power, SDPs have been studied extensively over the past few decades [1]}, [2]}, and have enable many applications in robust control [3]}, [4]}, trajectory planning [5]} and robotics [6]}. In theory, SDPs could be solved to arbitrary accuracy in polynomial time via efficient algorithms, such as second order interior point methods (IPMs). However, in practice, when the PSD cone, i.e., the set of all PSD matrices, is high dimensional, the solving time and memory requirements tend to be impractical [7]}. Over the past few years, extensive research has been performed with the aim of to improving the scalability of semidefinite programs [8]}. Since the computational cost of solving an SDP is primarily determined by the size of the largest semidefinite constraint, several methods have been proposed to decompose, or approximate high-dimensional PSD constraints into conic combinations of smaller PSD cones [9]}, [10]}.
